Notation

Rising and falling factorials

Write the rising factorial as

\[ x^\overline{n} = x(x+1)...(x+n-1) \]

and the falling factorial as

\[ x^{\frac{n}{}} = x(x-1)...(x-n+1) .\]

The position of the horizontal line immediately identifies the type of
factorial. The superscript reminds us these operations are akin to
exponentiation. Alternatives often employ parenthesess. Avoiding them here
reduces clutter and confusion.

Iverson brackets

We place a logical statement inside square brackets, and the whole expression
evaluates to 1 when the statement is true, and 0 otherwise. This is similar
to the logical and comparison operators of the C language.

For example,

\[ [x = 0] \]

is 1 when $x = 0$, and 0 otherwise (the impulse function). The Kronecker delta
$\delta_{i j}$ is simply

Stirling numbers

This is also the second subject of Knuth’s "Two Notes on Notation".

Writing $\binom{n}{k}$ for the number of ways of picking $k$ objects from a
set of size $n$ has triumphed over other schemes, such as an awkward concoction
involving a "C", a subscript and a superscript. Hopefully,

\[ \left[ {n \atop k} \right] \]

will one day be the standard notation for the number of permutations of $n$
objects with $k$ disjoint cycles, and

\[ \left\{ {n \atop k} \right\} \]

will be recoginzed as the number of ways to partition $n$ into $k$ subsets.

Incidentally, these are known as the Stirling numbers of the first and second kind.